Our results are a direct generalisation of those of [1]}, from ordered graphs to tournaments.
This generalisation is non-trivial: as explained in sec:proof-overview,
it is not possible to FO transduce orders from arbitrary tournaments—which would be the obvious way to reduce the problem from tournaments to ordered graphs.
On the other hand, there is a natural way to form a tournaments from an ordered graph \((G,<)\) :
\(x \rightarrow y\)  is an arc of \(T\)  if and only if \(x<y\)  and \(xy\)  is an edge of \(G\) , or \(y<x\)  and \(xy\)  is not an edge of \(G\) .
This gives a way to create tournaments with bounded twin-width:
starting with any graph \(G\)  with bounded twin-width,
consider a total order \(<\)  on its vertices witness of its twin-width, and interpret \(T\)  from \((G,<)\)  as before.
This paper can be seen as the reverse operation,
that is a way of producing an order \(<\)  from a tournament \(T\) .
